Publishing House A K A P I T
C
OMPUTER
M
ETHODS IN
M
ATERIALS
S
CIENCE
Informatyka w Technologii Materiałów
Vol. 9, 2009, No. 1
THERMO-MECHANICAL FORMING OF Al-Mg-Si SHEET
A.H.(TON) VAN DEN BOOGAARD1,SRIHARI KURUKURI2,MANOJIT GHOSH2,ALEXIS G.MIROUX2 1University of Twente, Enschede, The Netherlands
2Materials Innovation Institute, Delft, The Netherlands
Corresponding Author: a.h.vandenboogaard@utwente.nl (A.H. van den Boogaard)
Abstract
In warm forming of aluminum sheet, the temperature and strain rates vary considerably. In simulations, the material model must be capable to predict stresses within this wide range. Here, the physically based Nes model is used to describe the behavior of AA6061-T4 sheet material under warm forming conditions. A significant change of earing behavior is found between room temperature and 250 ºC. Crystal plasticity calculations showed a reasonable correspondence of changing r-values if extra slip systems are considered at high temperatures. Satisfactory results are obtained for simula-tion of tensile tests and cylindrical deep drawing.
Key words: warm forming, aluminum, material model, deep drawing, Nes model
1. INTRODUCTION
In recent years aluminum alloys are increasingly used in the automotive industry in order to reduce weight. However, aluminum alloys usually show lower room temperature formability, compared to mild steel. One way to overcome the poor formabil-ity of aluminum sheet is by temperature enhanced forming. In this process, parts of the tools are heated and other parts are cooled, in order to increase the formability. Forming of 5xxx series at elevated tem-peratures has been reported by many researchers [1-5]. Recently researchers are inclined to 6xxx series alloys with the intention to achieve improved form-ability at warm temperatures. The Al-Mg-Si alloys have good corrosion resistance, and obtain high strength, controlled by the precipitates formed dur-ing the agdur-ing treatment. Therefore, the age harden-ing response of these alloys is very significant and hence control of precipitation during thermo-mechanical treatment is critical for attaining optimal alloy performance. This process is industrially more
challenging and more complex in terms of micro-structure-mechanical behavior relationship.
Experience with temperature controlled forming processes is lacking and numerical models are bene-ficial in optimizing the forming processes. The ap-plicability of commonly applied constitutive models for aluminum is limited in terms of varying strain, strain rate, temperature and changing microstructure. Particularly in warm forming the strain rate and temperature influence cannot be ignored. Material models based on consideration of the underlying physical processes are expected to have a larger range of usability in this respect.
In cylindrical deep drawing, earing is a promi-nent indicator of anisotropy of sheet material. Ap-pearance of ears is directly related to undesirable effects like thickness variations. The formation of ears at room temperature has been widely investi-gated and the relation with the crystallographic tex-ture has been established. The earing behavior at elevated temperature is much less documented. In the present paper the relation between plastic
anisot-C OMPUTER M ETHODS IN M ATERIALS S CIENCE
ropy and crystallographic texture as a function of deformation temperature is investigated. Deep draw-ing experiments with AA6061 alloy are compared with numerical predictions.
2. EXPERIMENTS
Aluminum alloy AA6061 has been used for the present investigation. The material was cold rolled, solutionized and naturally aged (T4). The sheet thickness is 1.2 mm. The alloy contains 0.95 Mg, 0.62 Si, 0.2 Cu and 0.35 Fe in wt%.
Fig. 1. Dimensions of the tools for cylindrical cup deep drawing.
The sheet has been deep-drawn at room tempera-ture and at 250 ºC using a 1000 kN hydraulic press. Experiments were performed with a tool set of which the dimensions are given in figure 1. All ex-periments were performed with blanks of 220 mm. Blanks were drawn up to a depth of 64 mm with punch velocity 83 mm/min and blank holder pressure of 4.1 MPa at room temperature and with punch velocity 74 mm/min and blank holder pressure 2.5 MPa at 250 ºC. For warm deep drawing, the die and the blank holder were heated by heat rods while the punch was water cooled and kept below 30 ºC. The blank was heated up in contact with the die and was hold for 30 s at 250 ºC before being drawn. Drawing was stopped before the blank completely flowed inside the die. The cup was water quenched after warm drawing. The foot-print of the cups was meas-ured as a parameter of plastic anisotropy. The dis-tance between the cup central axis and the outer circumference was measured and plotted as a func-tion of the angle to the rolling direcfunc-tion.
Tensile tests were performed at room tempera-ture and 250 ºC with the thermo-mechanical simula-tor Gleeble 3800. Tensile specimens were taken at several angles to the rolling direction and deformed to 10% at a strain rate of 0.01 s-1 to measure r-values.
The texture of the as-received material has been
measured by X-ray diffraction while those of the deformed specimens have been measured by EBSD. The {200}, {220}, {111}, and {311} pole figures were measured and orientation distribution functions (ODFs) were calculated by the series expansion method, with the assumption of orthorhombic sam-ple symmetry.
Figure 2(a) shows that increasing the deforma-tion temperature from room temperature to 250 °C affects the normal anisotropy for all directions but does not change the r-value profile much. At both temperatures the r-value shows a minimum at 40– 50° to the rolling direction. Deformation temperature also strongly influences the earing profile of the drawn cups (figure 2(b)). At room temperature a 4-fold symmetry is observed with ears along RD and TD, while at 250 °C the earing profile exhibits a 2-fold symmetry although it shows a tendency to maintain a small local maximum along the rolling direction (0° and 180°).
a)
b)
Fig. 2. a) r-value measured by tensile tests (symbols) and
calcu-lated by VPSC model (lines). b) Drawn cup foot-print represent-ing the earrepresent-ing profile at room temperature and 250 ºC.
The orientation distribution function (ODF) of the AA6061-T4 sheet shows a texture dominated by Cube grains and an η-fibre (ϕ ϕ1= 2=0o) with
a weaker Goss component as shown in figure 3. Figure 4 shows the texture in the flange along the rolling direction after deep drawing at room tem-perature (RT) and at 250 ºC. It can be well noticed
0 0,3 0,6 0,9 1,2 0 15 30 45 60 75 90 Angle to RD r-v a lu e 72 74 76 78 80 82 0 45 90 135 180 225 270 315 360 Angle (degree) D ist. fr o m cen tr e (m m )
C OMPUTER M ETHODS IN M ATERIALS S CIENCE
that the texture changes after deformation while the textures obtained after drawing at both temperatures are qualitatively similar. For both textures the maximum has been found close to P with an orienta-tion spread along the α-fibre (φ=45 ,o ϕ2=0o) and
towards Cube. There is also an η-fibre between Goss and Cube. Hence it can be observed that tem-perature did not bring much change in the deep-drawing texture.
Fig. 3. ODF for AA 6061-T4 sheet calculated from XRD
meas-urement.
A visco-plastic self consistent model [6] was used to further investigate the effect of temperature on plastic deformation by assuming that temperature controls the critical resolved shear stresses (CRSS), that means the number of activated slip systems. At room temperature only the octahedral slip systems {111}<110> are allowed, while at 250 ºC four fami-lies of slip systems {111}<110>, {110}<110>, {100}<110> and {112}<110> are possible with respective CRSS ratio 0.9:1:1.1:1. The initial ex-perimental texture is discretized with 2000 orienta-tions. The modeled r-value curves with 1 and 4 slip systems respectively are shown in figure 2(a). The results at room temperature compare qualitatively well with the experimental measurements. The CRSS ratio of the four families of slip systems men-tioned above have been obtained by fitting the calcu-lated r-values to the r-values measured at 250 ºC. It is evident that by introducing more slip systems it is possible to increase the overall r-value and to obtain an r-value profile close to the experimental one. It
can be concluded that, although textures are similar at different temperatures, the activation of different families of slip systems directly affects the r-value profile.
3. MATERIAL MODEL
In this work, the standard model for plastic de-formation is used: a combination of a yield function to transform a 3-dimensional stress state to a scalar equivalent stress and a hardening model to deter-mine the size of the elastic domain by isotropic hardening. In the present study, the anisotropic yield function of Vegter [7] is used. For temperature and strain-rate sensitive work hardening, a physically based work hardening model by Nes [8] is used, in which the evolution of microstructure is defined by three internal state variables. The model developed by Nes and coworkers, has earlier been referred to as ALFLOW and MMP (Microstructure based Metal Plasticity) model but will here simply be referred to as the Nes model. The model approach relies on
a multi parameter description for the microstructure evolution by combining models for the dislocation storage problem with models for dynamic recovery of network dislocations and sub-boundary structures. The model also includes the thermal stress contribu-tions, static contributions from clusters and constitu-ents along with the contributions from precipitates. Hence the model would give an adequate description for stress-strain behavior at various strain rates and temperatures along with strain rate jumps. The uni-fied nature of the model is characterized in its gen-eral applicability, covering both low and high tem-perature properties of metals, in the form of a fully integrated work hardening description. Extensive presentations of the work hardening part of the model are given in [8,9,10].
3.1. Microstructure evolution
At small strains, the stored dislocations are ar-ranged in a cell structure characterized by a cell size,
Fig. 4. ODF in the flange after drawing (a) at RT and (b) at 250 ºC.
ϕ1
C OMPUTER M ETHODS IN M ATERIALS S CIENCE
δ, cell walls of thickness h and dislocation density b
ρ and a lower dislocation density within cells ρi. At large strains the cell walls have collapsed into sub-boundaries of a well defined misorientation ϕ as shown in figure 5. In the Nes model onlyρi, δ and ϕ are independent state variables. The micro-structure evolution is obtained by solving a set of differential equations describing the evolution of these parameters. Leaving out the details these can in principle be written as:
; i i i d d d d d d + − = ρ + ρ ρ γ γ γ ; d d d d d d − + = +
δ
δ
δ
γ
γ
γ
(1) d d d d d d + − = +ϕ
ϕ
ϕ
γ
γ
γ
Here,γis the resolved shear strain, which is de-fined as the algebraic sum of resolved shears of each slip system in the Taylor theory and interpreted as an average of the grains in this context.
i
dρ+ dγand dδ− dγ are storage terms, describing different
ways of athermal storage of dislocations, whereas
i
dρ− dγ describes dynamic recovery of cell interior
dislocations by dipole annihilation. dδ+ dγ
repre-sents subgrain growth at elevated temperatures. Modeling of the misorientation, dϕ γd , is currently based on a phenomenological approach. Explicit expressions for these terms can be found in [8,10].
(a) Small deformations (b) Large deformations
Fig. 5. Schematic presentation of the dislocation cell structure.
3.2. The flow stress
The flow stress, τ, at a constant microstructure is commonly defined in terms of a thermal stress
component, τt, and an athermal component, τa, so
that τ τ τ= +t a. The thermal component (also re-ferred to as the effective stress) is due to short range interactions between mobile dislocations and inter-secting stored ones, dragging of jogs, and elements in solid solution. The athermal component character-izes the rate- and temperature-independent interac-tion of dislocainterac-tions with long range barriers. In the treatments by Nes and co-workers [8-10], the stress required for dislocation migration is written as
t p cl 1 1 c i i q Gb⎡ ⎛ ⎞ = + + + ⎢ ⎜Γ ⎜ ⎟⎟ + ⎢ ⎝ ⎠ ⎣ τ τ τ τ α ρ δ ρ
( )
2 2 2 1 1 ˆ 0 c c i q q Gb D ⎤ ⎛ ⎞ ⎡ ⎤ Γ ⎜⎜ ⎟⎟ ⎥+ ⎢Γ + ⎥ ⎣ ⎦ ⎥ ⎝δ ρ ⎠ ⎦δ α δ (2)The Orowan stress τp is due to non-deformable particles. The clustering stress, τcl, due to clusters
formed by supersaturated alloying components with low diffusivity (Fe, Mn, Si). The α1-term represents the contribution from stored dislocations and the αˆ2
-term represents the contributionfrom subgrains and grain boundaries. G is the shear modulus, b is the Burger’s vector, δ and D are the cell/sub-grain size and grain size respectively. The functions Γ and 1 Γ2
are statistical distribution functions of sub-grain sizes within each grain.
In applications of the model the stress tensor at a macroscopic continuum scale is required repre-senting contributions from many grains of various crystallographic orientation and microstructure. In forming operations the texture changes are small and the anisotropy can be captured by the Vegter yield locus, providing a computational cost efficient ap-proach.
In figure 6, the simulated stress-strain curves are plotted for the Nes model, together with the experi-mental data. The stress-strain curves for tempera-tures of 25 ºC, 150 ºC and 250 ºC are plotted in figure 6(a) for a strain rate of =0.01s-1
&
ε and in
fig-ure 6(b) for a strain rate of =0.1s .-1
&
ε At higher
strain rate and at temperatures below 150 ºC the model performs quite well. For the higher tempera-tures, i.e., at 250 ºC the differences are slightly lar-ger. It can also be observed that at low strain rate and temperatures below 150 ºC the initial yield stress is underestimated with the Nes model.
In the experiment at 250 ºC the lower strain rate yields a slightly higher stress than the higher strain
C OMPUTER M ETHODS IN M ATERIALS S CIENCE
rate, which looks unusual. A possible explanation could be that at lower strain rate more time is avail-able for dynamic precipitation. However, as can be seen in figure 6 the Nes model cannot describe the effect of dynamic precipitation.
(a)
(b)
Fig. 6. True stress-strain curves – experiments and model.
4. SIMULATION OF CYLINDRICAL CUP DEEP DRAWING
The industrial relevance of the implemented ma-terial model is discussed in terms of a case study concerned with warm deep drawing of cylindrical cups. Orthotropic symmetry was assumed for the material model. A quarter of the blank was modeled and boundary conditions were applied on the dis-placement degrees of freedom to represent the sym-metry. The sheets were modeled with 998 discrete Kirchhoff triangular shell elements with 3 transla-tional, 3 rotational and 1 temperature degree of free-dom per node. The tools were modeled as rigid con-tours with a prescribed temperature. Simulations with the Vegter yield locus and the Nes hardening model implemented in the in-house implicit code DiekA are performed at various temperatures. The global convergence criterion was set to 0.5% relative unbalance force.
4.1. Friction
The friction between tool and work piece is one of the least known factors in the simulations. Fric-tion tests on an AA5754-O material, steel tools and a similar lubricant as used here at room temperature showed a friction coefficient of 0.06, which is lower than the value of 0.12 that is commonly used based on simulation experience. Experiments at high tem-peratures showed values varying between 0.12 and 0.18, both for a temperature of 175 ºC and 250 ºC. In the deep drawing process, the contact and friction conditions can be different than in the friction test, due to a locally different geometry and normal pres-sure.
To investigate the influence of the friction on the force–displacement curve and the thickness predic-tion, a parameter study was performed for a flange temperature of 175 ºC. In figure 7 the influence of friction on the punch force is shown for an analysis with the 1-parameter Bergström model [5]. In three analyses, the friction coefficient below 90 ºC, μl, was kept at 0.06, but the friction coefficient above 110 ºC, μh, was subsequently set to 0.06, 0.12 and 0.18. A linear interpolation was used between these temperatures. With a friction coefficient of 0.18 the calculated punch force–displacement diagram re-sembles the experimental one quite well. As a result of the increased friction, however, the predicted thickness strain deteriorates (see figure 7).
As an alternative parameter set, the friction coef-ficient for all tool blank contact was set at 0.12. This is compared with the original set with a friction co-efficient of 0.12 between die and blank and blank holder and blank and a friction coefficient of 0.06 between punch and blank. Interestingly, increasing the punch–blank friction has almost no effect on the punch force–displacement curve. The predicted thickness of the bottom area, however, improves considerably, while the predicted thickness in wall and flange hardly changes. This shows that the fric-tion condifric-tions are significant parameters in the simulation and the current uncertainties should be resolved. For the simulations in the next section the experimentally determined friction coefficients of 0.06 below 90 ºC and 0.12 above 110 ºC are used.
C OMPUTER M ETHODS IN M ATERIALS S CIENCE
Fig. 7. Influence of friction on calculated force and thickness.
4.2. Results
In figures 8 and 9, the force–displacement dia-grams of the punch and the thickness distributions of the cup at a depth of 64 mm are plotted for the ex-periments and the simulations. Comparing the dif-ferent punch force–displacement curves, it can be seen that the general trends with changing tempera-ture are predicted well. However, at 25 ºC the punch force is overestimated after highest point of the ex-perimental curve as shown in figure 8(a). At 250 ºC the punch force is underestimated after the highest point of experimental curve as shown in figure 8(b). The thickness distribution vs. distance from the cen-tre of the cup is shown in figure 9. At room tempera-ture a good agreement can be observed for the wall thickness but the thickness is underestimated in the bottom of the cup, see figure 9(a). At high tempera-tures the bottom thickness increases, while the wall thickness reduces compared to room temperature. It is observed that this trend is also represented by the model, but the effect is overestimated as shown in figure 9(b).
(a)
(b)
Fig. 8. Punch force-displacement curves from the Nes model
compared with experiments.
(a)
(b)
Fig. 9. Thickness distribution curves from the Nes model
C OMPUTER M ETHODS IN M ATERIALS S CIENCE ACKNOWLEDGEMENTS
This research was carried out under the project number MC1.02106 in the framework of the Re-search Program of the Materials innovation institute M2i (www.m2i.nl), the former Netherlands Institute for Metals Research. The financial support from Materials Innovation Institute, M2i, Netherlands, is highly acknowledged. The authors are indebted to P.J. Bolt and R. Werkhoven of TNO Science and Industry for performing the warm deep drawing experiments.
REFERENCES
1. Abedrabbo, N., Pourboghrat, F., Carsley, J., Forming of AA 5182-O and AA 5754-O at elevated temperatures using coupled thermo-mechanical finite element models, Int. J. of Plast., 23, 2007, 841–875.
2. Ayres, R., Alloying aluminum with magnesium for ductil-ity at warm temperatures (25 to 250 ºC), Metall. Trans. A, 10, 1979, 849–854.
3. Bolt, P.J., Lamboo, N.A.P.M., Rozier, P.J.C.M., Feasibility of warm Drawing of Aluminum Products, J. Mat. Proc. Tech., 115, 2001, 118–221.
4. Li, D., Ghosh, A.K., Biaxial warm forming behavior of aluminum sheet, J. Mat. Proc. Tech., 145, 2004, 281–293. 5. Van den Boogaard, A.H., Huétink, J., Simulation of
alumi-num sheet forming at elevated temperatures, Comp. Meth-ods. in App. Mech. and Engg., 195, 2006, 6691–6709. 6. Lebensohn, R.A., Tomé, C.N. A self-consistent anisotropic
approach for the simulation of plastic deformation and tex-ture development of polycrystals: Application to zirconium alloys, Acta Metall. Et Mater., 41(9), 1993, 2611–2624. 7. Vegter, H., Van den Boogaard, A.H., A plane stress yield
function for anisotropic sheet material by interpolation of biaxial stress states, Int. J. Plast., 2006, 22, 557–580. 8. Nes, E., Modeling of work hardening and stress saturation
in FCC metals, Prog. in Mat. Sci., 145, 1998, 129-193. 9. Holmedal, B., Marthinsen, K., Nes, E., A unified
micro-structural metal plasticity model applied in testing, process-ing, and forming of aluminum alloys, Z. Metallkd., 96, 2005, 532–545.
10. Nes, E., Marthinsen, K., Modeling the evolution in micro-structure and properties during plastic deformation of FCC-metals and alloys—an approach towards a unified model, Mat. Sci. and Engg. A, 322, 2002, 176–193.
TERMOMECHANICZNA OBRÓBKA PLASTYCZNA BLACH Al-Mg-Si
Streszczenie
Temperatura oraz rozkład prędkości odkształcenia podczas formowania aluminium na ciepło charakteryzują się duża nie-jednorodnością. Wykorzystywane do analizy modele naprężenia uplastyczniającego powinny uwzględniać wpływ tych niejedno-rodności. Opracowany w niniejszej pracy fizyczny model Nes wykorzystano do symulacji tłoczenia aluminium AA6061-T4 w warunkach odkształcenia na ciepło. W trakcie analizy zaobser-wowano znaczące różnice w kształcie wypływki uzyskanej podczas odkształcenia w temperaturze pokojowej oraz 250 ºC. Analiza z wykorzystaniem modelu plastyczności kryształów wykazała aktywności dodatkowych systemów poślizgu w pod-wyższonych temperaturach. W pracy przedstawiono wyniki analizy numerycznej dla procesów rozciągania oraz tłoczenia.
Submitted: October 1, 2008 Submitted in a revised form: November 3, 2008 Accepted: November 3, 2008
